On Yokoi's Invariants and the Ankeny-Artin-Chowla conjecture

Işlkay S., Peki˙n A.

International Journal of Number Theory, vol.18, no.3, pp.473-484, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 18 Issue: 3
  • Publication Date: 2022
  • Doi Number: 10.1142/s1793042122500270
  • Journal Name: International Journal of Number Theory
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Page Numbers: pp.473-484
  • Keywords: Continued fraction, class number, fundamental unit, real quadratic field, CLASS-NUMBERS, FUNDAMENTAL UNITS, BOUNDS
  • Kocaeli University Affiliated: No

Abstract

© 2022 World Scientific Publishing Company.Let d be a positive square-free integer and d = (Td + Udd)/2 > 1 be the fundamental unit of the real quadratic field ℚ(d). The Ankeny-Artin-Chowla (AAC) conjecture asserts that Up≢0 (mod p) for primes p 1 (mod 4), which still remains unsolved. In this paper, sufficient conditions for Ud < d have been given in terms of Yokoi's invariants nd and md, and it has been shown that the AAC conjecture is true in some special cases.